LMFDB - Embedded newform 84.16.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,16,Mod(1,84)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	84.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.862544284$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 47201410x^{2} - 158185874320x - 140304738691800 $ x^4 - 2x^3 - 47201410x^2 - 158185874320*x - 140304738691800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4150.04$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2187.00 q^{3} -47298.7 q^{5} -823543. q^{7} +4.78297e6 q^{9} +O(q^{10})$	$q+2187.00 q^{3} -47298.7 q^{5} -823543. q^{7} +4.78297e6 q^{9} +5.21658e7 q^{11} +5.45608e7 q^{13} -1.03442e8 q^{15} -9.14169e8 q^{17} +6.40186e9 q^{19} -1.80109e9 q^{21} -1.99380e10 q^{23} -2.82804e10 q^{25} +1.04604e10 q^{27} -4.28394e10 q^{29} +1.44532e11 q^{31} +1.14087e11 q^{33} +3.89525e10 q^{35} -4.69520e11 q^{37} +1.19324e11 q^{39} +9.48815e11 q^{41} -5.07039e11 q^{43} -2.26228e11 q^{45} +5.85306e12 q^{47} +6.78223e11 q^{49} -1.99929e12 q^{51} -5.53193e12 q^{53} -2.46738e12 q^{55} +1.40009e13 q^{57} +3.05101e12 q^{59} +3.52220e13 q^{61} -3.93898e12 q^{63} -2.58066e12 q^{65} -2.25619e13 q^{67} -4.36045e13 q^{69} +8.27529e13 q^{71} +1.37700e13 q^{73} -6.18493e13 q^{75} -4.29608e13 q^{77} +1.67933e13 q^{79} +2.28768e13 q^{81} +7.09963e13 q^{83} +4.32390e13 q^{85} -9.36897e13 q^{87} -5.21634e14 q^{89} -4.49332e13 q^{91} +3.16091e14 q^{93} -3.02800e14 q^{95} +9.15443e12 q^{97} +2.49508e14 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9}+O(q^{10}) $ 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9	\( 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9} - 61540668 q^{11} + 369006456 q^{13} + 602378532 q^{15} - 1458569484 q^{17} - 2691246184 q^{19} - 7204354164 q^{21} + 18028282212 q^{23} + 8448695516 q^{25} + 41841412812 q^{27} - 14387019264 q^{29} + 271402618248 q^{31} - 134589440916 q^{33} - 226833389748 q^{35} + 1719450823976 q^{37} + 807017119272 q^{39} + 2784989291868 q^{41} + 1897932676640 q^{43} + 1317401849484 q^{45} + 1872077464008 q^{47} + 2712892291396 q^{49} - 3189891461508 q^{51} + 4740642191736 q^{53} - 21671359654504 q^{55} - 5885755404408 q^{57} + 13984487218296 q^{59} + 32810552663568 q^{61} - 15755922556668 q^{63} + 183566971857672 q^{65} + 4158516498856 q^{67} + 39427853197644 q^{69} + 70250046828444 q^{71} + 27064251481456 q^{73} + 18477297093492 q^{75} + 50681386346724 q^{77} - 238381767283992 q^{79} + 91507169819844 q^{81} - 233193205883808 q^{83} - 893307076687696 q^{85} - 31464411130368 q^{87} + 316154161239228 q^{89} - 303892683793608 q^{91} + 593557526108376 q^{93} - 11\!\cdots\!76 q^{95}+ \cdots - 294347107283292 q^{99}+O(q^{100}) \) 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9 - 61540668 * q^11 + 369006456 * q^13 + 602378532 * q^15 - 1458569484 * q^17 - 2691246184 * q^19 - 7204354164 * q^21 + 18028282212 * q^23 + 8448695516 * q^25 + 41841412812 * q^27 - 14387019264 * q^29 + 271402618248 * q^31 - 134589440916 * q^33 - 226833389748 * q^35 + 1719450823976 * q^37 + 807017119272 * q^39 + 2784989291868 * q^41 + 1897932676640 * q^43 + 1317401849484 * q^45 + 1872077464008 * q^47 + 2712892291396 * q^49 - 3189891461508 * q^51 + 4740642191736 * q^53 - 21671359654504 * q^55 - 5885755404408 * q^57 + 13984487218296 * q^59 + 32810552663568 * q^61 - 15755922556668 * q^63 + 183566971857672 * q^65 + 4158516498856 * q^67 + 39427853197644 * q^69 + 70250046828444 * q^71 + 27064251481456 * q^73 + 18477297093492 * q^75 + 50681386346724 * q^77 - 238381767283992 * q^79 + 91507169819844 * q^81 - 233193205883808 * q^83 - 893307076687696 * q^85 - 31464411130368 * q^87 + 316154161239228 * q^89 - 303892683793608 * q^91 + 593557526108376 * q^93 - 1197067113655776 * q^95 + 529302182472448 * q^97 - 294347107283292 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	0	0
$5$	−47298.7	−0.270754	−0.135377	−	0.990794i	$-0.543225\pi$
$5$	−47298.7	−0.270754	−0.135377	+	0.990794i	$0.543225\pi$
$6$	0	0
$7$	−823543.	−0.377964
$7$	−823543.	−0.377964
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	5.21658e7	0.807126	0.403563	−	0.914952i	$-0.367772\pi$
$11$	5.21658e7	0.807126	0.403563	+	0.914952i	$0.367772\pi$
$12$	0	0
$13$	5.45608e7	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$13$	5.45608e7	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$14$	0	0
$15$	−1.03442e8	−0.156320
$16$	0	0
$17$	−9.14169e8	−0.540331	−0.270165	−	0.962814i	$-0.587078\pi$
$17$	−9.14169e8	−0.540331	−0.270165	+	0.962814i	$0.587078\pi$
$18$	0	0
$19$	6.40186e9	1.64306	0.821531	−	0.570164i	$-0.193120\pi$
$19$	6.40186e9	1.64306	0.821531	+	0.570164i	$0.193120\pi$
$20$	0	0
$21$	−1.80109e9	−0.218218
$22$	0	0
$23$	−1.99380e10	−1.22102	−0.610511	−	0.792008i	$-0.709036\pi$
$23$	−1.99380e10	−1.22102	−0.610511	+	0.792008i	$0.709036\pi$
$24$	0	0
$25$	−2.82804e10	−0.926692
$26$	0	0
$27$	1.04604e10	0.192450
$28$	0	0
$29$	−4.28394e10	−0.461167	−0.230583	−	0.973053i	$-0.574063\pi$
$29$	−4.28394e10	−0.461167	−0.230583	+	0.973053i	$0.574063\pi$
$30$	0	0
$31$	1.44532e11	0.943519	0.471759	−	0.881727i	$-0.343619\pi$
$31$	1.44532e11	0.943519	0.471759	+	0.881727i	$0.343619\pi$
$32$	0	0
$33$	1.14087e11	0.465994
$34$	0	0
$35$	3.89525e10	0.102335
$36$	0	0
$37$	−4.69520e11	−0.813095	−0.406547	−	0.913630i	$-0.633267\pi$
$37$	−4.69520e11	−0.813095	−0.406547	+	0.913630i	$0.633267\pi$
$38$	0	0
$39$	1.19324e11	0.139234
$40$	0	0
$41$	9.48815e11	0.760856	0.380428	−	0.924811i	$-0.375777\pi$
$41$	9.48815e11	0.760856	0.380428	+	0.924811i	$0.375777\pi$
$42$	0	0
$43$	−5.07039e11	−0.284465	−0.142232	−	0.989833i	$-0.545428\pi$
$43$	−5.07039e11	−0.284465	−0.142232	+	0.989833i	$0.545428\pi$
$44$	0	0
$45$	−2.26228e11	−0.0902512
$46$	0	0
$47$	5.85306e12	1.68519	0.842595	−	0.538547i	$-0.181027\pi$
$47$	5.85306e12	1.68519	0.842595	+	0.538547i	$0.181027\pi$
$48$	0	0
$49$	6.78223e11	0.142857
$50$	0	0
$51$	−1.99929e12	−0.311960
$52$	0	0
$53$	−5.53193e12	−0.646856	−0.323428	−	0.946253i	$-0.604835\pi$
$53$	−5.53193e12	−0.646856	−0.323428	+	0.946253i	$0.604835\pi$
$54$	0	0
$55$	−2.46738e12	−0.218532
$56$	0	0
$57$	1.40009e13	0.948623
$58$	0	0
$59$	3.05101e12	0.159608	0.0798039	−	0.996811i	$-0.474571\pi$
$59$	3.05101e12	0.159608	0.0798039	+	0.996811i	$0.474571\pi$
$60$	0	0
$61$	3.52220e13	1.43496	0.717480	−	0.696579i	$-0.245295\pi$
$61$	3.52220e13	1.43496	0.717480	+	0.696579i	$0.245295\pi$
$62$	0	0
$63$	−3.93898e12	−0.125988
$64$	0	0
$65$	−2.58066e12	−0.0652950
$66$	0	0
$67$	−2.25619e13	−0.454795	−0.227397	−	0.973802i	$-0.573022\pi$
$67$	−2.25619e13	−0.454795	−0.227397	+	0.973802i	$0.573022\pi$
$68$	0	0
$69$	−4.36045e13	−0.704958
$70$	0	0
$71$	8.27529e13	1.07981	0.539903	−	0.841727i	$-0.318461\pi$
$71$	8.27529e13	1.07981	0.539903	+	0.841727i	$0.318461\pi$
$72$	0	0
$73$	1.37700e13	0.145886	0.0729430	−	0.997336i	$-0.476761\pi$
$73$	1.37700e13	0.145886	0.0729430	+	0.997336i	$0.476761\pi$
$74$	0	0
$75$	−6.18493e13	−0.535026
$76$	0	0
$77$	−4.29608e13	−0.305065
$78$	0	0
$79$	1.67933e13	0.0983859	0.0491930	−	0.998789i	$-0.484335\pi$
$79$	1.67933e13	0.0983859	0.0491930	+	0.998789i	$0.484335\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	7.09963e13	0.287177	0.143589	−	0.989637i	$-0.454136\pi$
$83$	7.09963e13	0.287177	0.143589	+	0.989637i	$0.454136\pi$
$84$	0	0
$85$	4.32390e13	0.146296
$86$	0	0
$87$	−9.36897e13	−0.266255
$88$	0	0
$89$	−5.21634e14	−1.25009	−0.625044	−	0.780589i	$-0.714919\pi$
$89$	−5.21634e14	−1.25009	−0.625044	+	0.780589i	$0.714919\pi$
$90$	0	0
$91$	−4.49332e13	−0.0911500
$92$	0	0
$93$	3.16091e14	0.544741
$94$	0	0
$95$	−3.02800e14	−0.444865
$96$	0	0
$97$	9.15443e12	0.0115039	0.00575193	−	0.999983i	$-0.498169\pi$
$97$	9.15443e12	0.0115039	0.00575193	+	0.999983i	$0.498169\pi$
$98$	0	0
$99$	2.49508e14	0.269042
$100$	0	0
$101$	1.44518e15	1.34125	0.670627	−	0.741794i	$-0.266025\pi$
$101$	1.44518e15	1.34125	0.670627	+	0.741794i	$0.266025\pi$
$102$	0	0
$103$	−5.20282e14	−0.416831	−0.208415	−	0.978040i	$-0.566831\pi$
$103$	−5.20282e14	−0.416831	−0.208415	+	0.978040i	$0.566831\pi$
$104$	0	0
$105$	8.51892e13	0.0590833
$106$	0	0
$107$	9.43387e14	0.567952	0.283976	−	0.958831i	$-0.408346\pi$
$107$	9.43387e14	0.567952	0.283976	+	0.958831i	$0.408346\pi$
$108$	0	0
$109$	3.00488e15	1.57445	0.787224	−	0.616667i	$-0.211517\pi$
$109$	3.00488e15	1.57445	0.787224	+	0.616667i	$0.211517\pi$
$110$	0	0
$111$	−1.02684e15	−0.469441
$112$	0	0
$113$	−1.09257e15	−0.436878	−0.218439	−	0.975851i	$-0.570096\pi$
$113$	−1.09257e15	−0.436878	−0.218439	+	0.975851i	$0.570096\pi$
$114$	0	0
$115$	9.43044e14	0.330596
$116$	0	0
$117$	2.60963e14	0.0803867
$118$	0	0
$119$	7.52858e14	0.204226
$120$	0	0
$121$	−1.45597e15	−0.348548
$122$	0	0
$123$	2.07506e15	0.439280
$124$	0	0
$125$	2.78107e15	0.521659
$126$	0	0
$127$	1.05226e16	1.75224	0.876121	−	0.482091i	$-0.160123\pi$
$127$	1.05226e16	1.75224	0.876121	+	0.482091i	$0.160123\pi$
$128$	0	0
$129$	−1.10889e15	−0.164236
$130$	0	0
$131$	8.90971e15	1.17579	0.587894	−	0.808938i	$-0.299957\pi$
$131$	8.90971e15	1.17579	0.587894	+	0.808938i	$0.299957\pi$
$132$	0	0
$133$	−5.27221e15	−0.621019
$134$	0	0
$135$	−4.94761e14	−0.0521066
$136$	0	0
$137$	−7.04650e15	−0.664614	−0.332307	−	0.943171i	$-0.607827\pi$
$137$	−7.04650e15	−0.664614	−0.332307	+	0.943171i	$0.607827\pi$
$138$	0	0
$139$	1.73081e16	1.46433	0.732163	−	0.681130i	$-0.238511\pi$
$139$	1.73081e16	1.46433	0.732163	+	0.681130i	$0.238511\pi$
$140$	0	0
$141$	1.28006e16	0.972945
$142$	0	0
$143$	2.84621e15	0.194647
$144$	0	0
$145$	2.02625e15	0.124863
$146$	0	0
$147$	1.48327e15	0.0824786
$148$	0	0
$149$	3.75538e16	1.88693	0.943467	−	0.331465i	$-0.107543\pi$
$149$	3.75538e16	1.88693	0.943467	+	0.331465i	$0.107543\pi$
$150$	0	0
$151$	1.78033e16	0.809417	0.404708	−	0.914446i	$-0.367373\pi$
$151$	1.78033e16	0.809417	0.404708	+	0.914446i	$0.367373\pi$
$152$	0	0
$153$	−4.37244e15	−0.180110
$154$	0	0
$155$	−6.83617e15	−0.255461
$156$	0	0
$157$	−3.23219e15	−0.109711	−0.0548554	−	0.998494i	$-0.517470\pi$
$157$	−3.23219e15	−0.109711	−0.0548554	+	0.998494i	$0.517470\pi$
$158$	0	0
$159$	−1.20983e16	−0.373462
$160$	0	0
$161$	1.64198e16	0.461503
$162$	0	0
$163$	1.34083e16	0.343531	0.171766	−	0.985138i	$-0.445053\pi$
$163$	1.34083e16	0.343531	0.171766	+	0.985138i	$0.445053\pi$
$164$	0	0
$165$	−5.39615e15	−0.126170
$166$	0	0
$167$	1.21127e16	0.258742	0.129371	−	0.991596i	$-0.458704\pi$
$167$	1.21127e16	0.258742	0.129371	+	0.991596i	$0.458704\pi$
$168$	0	0
$169$	−4.82090e16	−0.941842
$170$	0	0
$171$	3.06199e16	0.547687
$172$	0	0
$173$	8.24389e16	1.35141	0.675704	−	0.737173i	$-0.263840\pi$
$173$	8.24389e16	1.35141	0.675704	+	0.737173i	$0.263840\pi$
$174$	0	0
$175$	2.32901e16	0.350257
$176$	0	0
$177$	6.67257e15	0.0921496
$178$	0	0
$179$	4.01334e16	0.509458	0.254729	−	0.967013i	$-0.418014\pi$
$179$	4.01334e16	0.509458	0.254729	+	0.967013i	$0.418014\pi$
$180$	0	0
$181$	4.65148e16	0.543253	0.271626	−	0.962403i	$-0.412438\pi$
$181$	4.65148e16	0.543253	0.271626	+	0.962403i	$0.412438\pi$
$182$	0	0
$183$	7.70305e16	0.828475
$184$	0	0
$185$	2.22077e16	0.220148
$186$	0	0
$187$	−4.76884e16	−0.436115
$188$	0	0
$189$	−8.61455e15	−0.0727393
$190$	0	0
$191$	4.77369e16	0.372481	0.186240	−	0.982504i	$-0.440370\pi$
$191$	4.77369e16	0.372481	0.186240	+	0.982504i	$0.440370\pi$
$192$	0	0
$193$	2.23373e17	1.61195	0.805973	−	0.591952i	$-0.201642\pi$
$193$	2.23373e17	1.61195	0.805973	+	0.591952i	$0.201642\pi$
$194$	0	0
$195$	−5.64389e15	−0.0376981
$196$	0	0
$197$	−2.66902e17	−1.65141	−0.825706	−	0.564101i	$-0.809223\pi$
$197$	−2.66902e17	−1.65141	−0.825706	+	0.564101i	$0.809223\pi$
$198$	0	0
$199$	1.85506e17	1.06404	0.532021	−	0.846731i	$-0.321433\pi$
$199$	1.85506e17	1.06404	0.532021	+	0.846731i	$0.321433\pi$
$200$	0	0
$201$	−4.93429e16	−0.262576
$202$	0	0
$203$	3.52801e16	0.174305
$204$	0	0
$205$	−4.48777e16	−0.206004
$206$	0	0
$207$	−9.53630e16	−0.407008
$208$	0	0
$209$	3.33958e17	1.32616
$210$	0	0
$211$	4.71501e17	1.74327	0.871633	−	0.490158i	$-0.163061\pi$
$211$	4.71501e17	1.74327	0.871633	+	0.490158i	$0.163061\pi$
$212$	0	0
$213$	1.80981e17	0.623426
$214$	0	0
$215$	2.39823e16	0.0770198
$216$	0	0
$217$	−1.19028e17	−0.356617
$218$	0	0
$219$	3.01150e16	0.0842273
$220$	0	0
$221$	−4.98778e16	−0.130306
$222$	0	0
$223$	−5.40716e17	−1.32033	−0.660165	−	0.751120i	$-0.729514\pi$
$223$	−5.40716e17	−1.32033	−0.660165	+	0.751120i	$0.729514\pi$
$224$	0	0
$225$	−1.35264e17	−0.308897
$226$	0	0
$227$	−2.30239e17	−0.492023	−0.246012	−	0.969267i	$-0.579120\pi$
$227$	−2.30239e17	−0.492023	−0.246012	+	0.969267i	$0.579120\pi$
$228$	0	0
$229$	1.66219e14	0.000332594 0	0.000166297	−	1.00000i	$-0.499947\pi$
$229$	1.66219e14	0.000332594 0	0.000166297		1.00000i	$0.499947\pi$
$230$	0	0
$231$	−9.39553e16	−0.176129
$232$	0	0
$233$	−6.46270e17	−1.13565	−0.567825	−	0.823149i	$-0.692215\pi$
$233$	−6.46270e17	−1.13565	−0.567825	+	0.823149i	$0.692215\pi$
$234$	0	0
$235$	−2.76842e17	−0.456272
$236$	0	0
$237$	3.67270e16	0.0568031
$238$	0	0
$239$	1.83413e17	0.266346	0.133173	−	0.991093i	$-0.457483\pi$
$239$	1.83413e17	0.266346	0.133173	+	0.991093i	$0.457483\pi$
$240$	0	0
$241$	−9.21604e17	−1.25723	−0.628617	−	0.777715i	$-0.716379\pi$
$241$	−9.21604e17	−1.25723	−0.628617	+	0.777715i	$0.716379\pi$
$242$	0	0
$243$	5.00315e16	0.0641500
$244$	0	0
$245$	−3.20791e16	−0.0386791
$246$	0	0
$247$	3.49291e17	0.396241
$248$	0	0
$249$	1.55269e17	0.165802
$250$	0	0
$251$	−8.29366e17	−0.834053	−0.417026	−	0.908894i	$-0.636928\pi$
$251$	−8.29366e17	−0.834053	−0.417026	+	0.908894i	$0.636928\pi$
$252$	0	0
$253$	−1.04008e18	−0.985519
$254$	0	0
$255$	9.45637e16	0.0844643
$256$	0	0
$257$	−2.07919e18	−1.75144	−0.875720	−	0.482819i	$-0.839613\pi$
$257$	−2.07919e18	−1.75144	−0.875720	+	0.482819i	$0.839613\pi$
$258$	0	0
$259$	3.86670e17	0.307321
$260$	0	0
$261$	−2.04899e17	−0.153722
$262$	0	0
$263$	−1.36831e17	−0.0969430	−0.0484715	−	0.998825i	$-0.515435\pi$
$263$	−1.36831e17	−0.0969430	−0.0484715	+	0.998825i	$0.515435\pi$
$264$	0	0
$265$	2.61653e17	0.175139
$266$	0	0
$267$	−1.14081e18	−0.721739
$268$	0	0
$269$	2.00989e18	1.20235	0.601174	−	0.799119i	$-0.294700\pi$
$269$	2.00989e18	1.20235	0.601174	+	0.799119i	$0.294700\pi$
$270$	0	0
$271$	−1.23645e18	−0.699694	−0.349847	−	0.936807i	$-0.613766\pi$
$271$	−1.23645e18	−0.699694	−0.349847	+	0.936807i	$0.613766\pi$
$272$	0	0
$273$	−9.82688e16	−0.0526255
$274$	0	0
$275$	−1.47527e18	−0.747957
$276$	0	0
$277$	1.13073e18	0.542950	0.271475	−	0.962446i	$-0.412489\pi$
$277$	1.13073e18	0.542950	0.271475	+	0.962446i	$0.412489\pi$
$278$	0	0
$279$	6.91291e17	0.314506
$280$	0	0
$281$	−5.92081e17	−0.255320	−0.127660	−	0.991818i	$-0.540747\pi$
$281$	−5.92081e17	−0.255320	−0.127660	+	0.991818i	$0.540747\pi$
$282$	0	0
$283$	−2.67818e18	−1.09507	−0.547534	−	0.836783i	$-0.684433\pi$
$283$	−2.67818e18	−1.09507	−0.547534	+	0.836783i	$0.684433\pi$
$284$	0	0
$285$	−6.62223e17	−0.256843
$286$	0	0
$287$	−7.81390e17	−0.287576
$288$	0	0
$289$	−2.02672e18	−0.708043
$290$	0	0
$291$	2.00207e16	0.00664175
$292$	0	0
$293$	−1.93321e18	−0.609216	−0.304608	−	0.952478i	$-0.598526\pi$
$293$	−1.93321e18	−0.609216	−0.304608	+	0.952478i	$0.598526\pi$
$294$	0	0
$295$	−1.44309e17	−0.0432144
$296$	0	0
$297$	5.45673e17	0.155331
$298$	0	0
$299$	−1.08784e18	−0.294462
$300$	0	0
$301$	4.17569e17	0.107517
$302$	0	0
$303$	3.16061e18	0.774374
$304$	0	0
$305$	−1.66595e18	−0.388521
$306$	0	0
$307$	8.29911e18	1.84287	0.921433	−	0.388538i	$-0.127020\pi$
$307$	8.29911e18	1.84287	0.921433	+	0.388538i	$0.127020\pi$
$308$	0	0
$309$	−1.13786e18	−0.240657
$310$	0	0
$311$	−1.22918e18	−0.247692	−0.123846	−	0.992301i	$-0.539523\pi$
$311$	−1.22918e18	−0.247692	−0.123846	+	0.992301i	$0.539523\pi$
$312$	0	0
$313$	5.37393e17	0.103207	0.0516035	−	0.998668i	$-0.483567\pi$
$313$	5.37393e17	0.103207	0.0516035	+	0.998668i	$0.483567\pi$
$314$	0	0
$315$	1.86309e17	0.0341118
$316$	0	0
$317$	2.33751e18	0.408140	0.204070	−	0.978956i	$-0.434583\pi$
$317$	2.33751e18	0.408140	0.204070	+	0.978956i	$0.434583\pi$
$318$	0	0
$319$	−2.23475e18	−0.372220
$320$	0	0
$321$	2.06319e18	0.327907
$322$	0	0
$323$	−5.85238e18	−0.887797
$324$	0	0
$325$	−1.54300e18	−0.223481
$326$	0	0
$327$	6.57167e18	0.909008
$328$	0	0
$329$	−4.82025e18	−0.636942
$330$	0	0
$331$	−6.16166e18	−0.778015	−0.389008	−	0.921235i	$-0.627182\pi$
$331$	−6.16166e18	−0.778015	−0.389008	+	0.921235i	$0.627182\pi$
$332$	0	0
$333$	−2.24570e18	−0.271032
$334$	0	0
$335$	1.06715e18	0.123137
$336$	0	0
$337$	1.23765e18	0.136576	0.0682881	−	0.997666i	$-0.478246\pi$
$337$	1.23765e18	0.136576	0.0682881	+	0.997666i	$0.478246\pi$
$338$	0	0
$339$	−2.38944e18	−0.252231
$340$	0	0
$341$	7.53962e18	0.761538
$342$	0	0
$343$	−5.58546e17	−0.0539949
$344$	0	0
$345$	2.06244e18	0.190870
$346$	0	0
$347$	2.60254e18	0.230636	0.115318	−	0.993329i	$-0.463211\pi$
$347$	2.60254e18	0.230636	0.115318	+	0.993329i	$0.463211\pi$
$348$	0	0
$349$	1.31991e19	1.12035	0.560176	−	0.828374i	$-0.310734\pi$
$349$	1.31991e19	1.12035	0.560176	+	0.828374i	$0.310734\pi$
$350$	0	0
$351$	5.70725e17	0.0464113
$352$	0	0
$353$	4.58364e18	0.357191	0.178596	−	0.983923i	$-0.442845\pi$
$353$	4.58364e18	0.357191	0.178596	+	0.983923i	$0.442845\pi$
$354$	0	0
$355$	−3.91411e18	−0.292361
$356$	0	0
$357$	1.64650e18	0.117910
$358$	0	0
$359$	−1.70404e19	−1.17023	−0.585115	−	0.810950i	$-0.698951\pi$
$359$	−1.70404e19	−1.17023	−0.585115	+	0.810950i	$0.698951\pi$
$360$	0	0
$361$	2.58027e19	1.69965
$362$	0	0
$363$	−3.18421e18	−0.201234
$364$	0	0
$365$	−6.51305e17	−0.0394991
$366$	0	0
$367$	2.10183e19	1.22349	0.611746	−	0.791054i	$-0.290467\pi$
$367$	2.10183e19	1.22349	0.611746	+	0.791054i	$0.290467\pi$
$368$	0	0
$369$	4.53815e18	0.253619
$370$	0	0
$371$	4.55578e18	0.244488
$372$	0	0
$373$	4.18439e18	0.215683	0.107841	−	0.994168i	$-0.465606\pi$
$373$	4.18439e18	0.215683	0.107841	+	0.994168i	$0.465606\pi$
$374$	0	0
$375$	6.08220e18	0.301180
$376$	0	0
$377$	−2.33735e18	−0.111215
$378$	0	0
$379$	1.98193e19	0.906348	0.453174	−	0.891422i	$-0.350292\pi$
$379$	1.98193e19	0.906348	0.453174	+	0.891422i	$0.350292\pi$
$380$	0	0
$381$	2.30129e19	1.01166
$382$	0	0
$383$	−2.18964e19	−0.925511	−0.462755	−	0.886486i	$-0.653139\pi$
$383$	−2.18964e19	−0.925511	−0.462755	+	0.886486i	$0.653139\pi$
$384$	0	0
$385$	2.03199e18	0.0825974
$386$	0	0
$387$	−2.42515e18	−0.0948215
$388$	0	0
$389$	8.53409e18	0.321022	0.160511	−	0.987034i	$-0.448686\pi$
$389$	8.53409e18	0.321022	0.160511	+	0.987034i	$0.448686\pi$
$390$	0	0
$391$	1.82267e19	0.659756
$392$	0	0
$393$	1.94855e19	0.678842
$394$	0	0
$395$	−7.94302e17	−0.0266383
$396$	0	0
$397$	7.66965e18	0.247655	0.123827	−	0.992304i	$-0.460483\pi$
$397$	7.66965e18	0.247655	0.123827	+	0.992304i	$0.460483\pi$
$398$	0	0
$399$	−1.15303e19	−0.358546
$400$	0	0
$401$	1.35101e19	0.404647	0.202324	−	0.979319i	$-0.435151\pi$
$401$	1.35101e19	0.404647	0.202324	+	0.979319i	$0.435151\pi$
$402$	0	0
$403$	7.88577e18	0.227539
$404$	0	0
$405$	−1.08204e18	−0.0300837
$406$	0	0
$407$	−2.44929e19	−0.656270
$408$	0	0
$409$	−4.12065e18	−0.106424	−0.0532122	−	0.998583i	$-0.516946\pi$
$409$	−4.12065e18	−0.106424	−0.0532122	+	0.998583i	$0.516946\pi$
$410$	0	0
$411$	−1.54107e19	−0.383715
$412$	0	0
$413$	−2.51264e18	−0.0603261
$414$	0	0
$415$	−3.35803e18	−0.0777542
$416$	0	0
$417$	3.78527e19	0.845429
$418$	0	0
$419$	7.08353e19	1.52632	0.763159	−	0.646211i	$-0.223647\pi$
$419$	7.08353e19	1.52632	0.763159	+	0.646211i	$0.223647\pi$
$420$	0	0
$421$	1.72106e19	0.357833	0.178917	−	0.983864i	$-0.442741\pi$
$421$	1.72106e19	0.357833	0.178917	+	0.983864i	$0.442741\pi$
$422$	0	0
$423$	2.79950e19	0.561730
$424$	0	0
$425$	2.58531e19	0.500720
$426$	0	0
$427$	−2.90068e19	−0.542364
$428$	0	0
$429$	6.22466e18	0.112379
$430$	0	0
$431$	−8.26903e19	−1.44170	−0.720850	−	0.693091i	$-0.756248\pi$
$431$	−8.26903e19	−1.44170	−0.720850	+	0.693091i	$0.756248\pi$
$432$	0	0
$433$	−3.87541e16	−0.000652616 0	−0.000326308	−	1.00000i	$-0.500104\pi$
$433$	−3.87541e16	−0.000652616 0	−0.000326308		1.00000i	$0.500104\pi$
$434$	0	0
$435$	4.43140e18	0.0720895
$436$	0	0
$437$	−1.27641e20	−2.00622
$438$	0	0
$439$	4.29194e19	0.651883	0.325941	−	0.945390i	$-0.394319\pi$
$439$	4.29194e19	0.651883	0.325941	+	0.945390i	$0.394319\pi$
$440$	0	0
$441$	3.24392e18	0.0476190
$442$	0	0
$443$	−1.03568e20	−1.46959	−0.734796	−	0.678288i	$-0.762722\pi$
$443$	−1.03568e20	−1.46959	−0.734796	+	0.678288i	$0.762722\pi$
$444$	0	0
$445$	2.46726e19	0.338466
$446$	0	0
$447$	8.21302e19	1.08942
$448$	0	0
$449$	−1.28299e20	−1.64580	−0.822899	−	0.568187i	$-0.807645\pi$
$449$	−1.28299e20	−1.64580	−0.822899	+	0.568187i	$0.807645\pi$
$450$	0	0
$451$	4.94957e19	0.614106
$452$	0	0
$453$	3.89357e19	0.467317
$454$	0	0
$455$	2.12528e18	0.0246792
$456$	0	0
$457$	−2.60255e19	−0.292434	−0.146217	−	0.989253i	$-0.546710\pi$
$457$	−2.60255e19	−0.292434	−0.146217	+	0.989253i	$0.546710\pi$
$458$	0	0
$459$	−9.56253e18	−0.103987
$460$	0	0
$461$	−2.70955e19	−0.285194	−0.142597	−	0.989781i	$-0.545545\pi$
$461$	−2.70955e19	−0.285194	−0.142597	+	0.989781i	$0.545545\pi$
$462$	0	0
$463$	−2.60790e19	−0.265726	−0.132863	−	0.991134i	$-0.542417\pi$
$463$	−2.60790e19	−0.265726	−0.132863	+	0.991134i	$0.542417\pi$
$464$	0	0
$465$	−1.49507e19	−0.147491
$466$	0	0
$467$	−1.68853e20	−1.61300	−0.806498	−	0.591237i	$-0.798640\pi$
$467$	−1.68853e20	−1.61300	−0.806498	+	0.591237i	$0.798640\pi$
$468$	0	0
$469$	1.85807e19	0.171896
$470$	0	0
$471$	−7.06880e18	−0.0633416
$472$	0	0
$473$	−2.64501e19	−0.229599
$474$	0	0
$475$	−1.81047e20	−1.52261
$476$	0	0
$477$	−2.64590e19	−0.215619
$478$	0	0
$479$	−2.63718e19	−0.208268	−0.104134	−	0.994563i	$-0.533207\pi$
$479$	−2.63718e19	−0.208268	−0.104134	+	0.994563i	$0.533207\pi$
$480$	0	0
$481$	−2.56174e19	−0.196086
$482$	0	0
$483$	3.59102e19	0.266449
$484$	0	0
$485$	−4.32993e17	−0.00311471
$486$	0	0
$487$	−5.70070e19	−0.397613	−0.198807	−	0.980039i	$-0.563707\pi$
$487$	−5.70070e19	−0.397613	−0.198807	+	0.980039i	$0.563707\pi$
$488$	0	0
$489$	2.93239e19	0.198338
$490$	0	0
$491$	−9.90233e19	−0.649570	−0.324785	−	0.945788i	$-0.605292\pi$
$491$	−9.90233e19	−0.649570	−0.324785	+	0.945788i	$0.605292\pi$
$492$	0	0
$493$	3.91624e19	0.249183
$494$	0	0
$495$	−1.18014e19	−0.0728441
$496$	0	0
$497$	−6.81506e19	−0.408128
$498$	0	0
$499$	1.83179e20	1.06444	0.532222	−	0.846605i	$-0.321357\pi$
$499$	1.83179e20	1.06444	0.532222	+	0.846605i	$0.321357\pi$
$500$	0	0
$501$	2.64904e19	0.149385
$502$	0	0
$503$	−3.08633e20	−1.68920	−0.844601	−	0.535396i	$-0.820163\pi$
$503$	−3.08633e20	−1.68920	−0.844601	+	0.535396i	$0.820163\pi$
$504$	0	0
$505$	−6.83551e19	−0.363150
$506$	0	0
$507$	−1.05433e20	−0.543773
$508$	0	0
$509$	−1.27050e20	−0.636199	−0.318099	−	0.948057i	$-0.603045\pi$
$509$	−1.27050e20	−0.636199	−0.318099	+	0.948057i	$0.603045\pi$
$510$	0	0
$511$	−1.13402e19	−0.0551397
$512$	0	0
$513$	6.69657e19	0.316208
$514$	0	0
$515$	2.46087e19	0.112858
$516$	0	0
$517$	3.05330e20	1.36016
$518$	0	0
$519$	1.80294e20	0.780235
$520$	0	0
$521$	−3.92772e19	−0.165142	−0.0825711	−	0.996585i	$-0.526313\pi$
$521$	−3.92772e19	−0.165142	−0.0825711	+	0.996585i	$0.526313\pi$
$522$	0	0
$523$	−1.23840e19	−0.0505938	−0.0252969	−	0.999680i	$-0.508053\pi$
$523$	−1.23840e19	−0.0505938	−0.0252969	+	0.999680i	$0.508053\pi$
$524$	0	0
$525$	5.09355e19	0.202221
$526$	0	0
$527$	−1.32126e20	−0.509812
$528$	0	0
$529$	1.30890e20	0.490896
$530$	0	0
$531$	1.45929e19	0.0532026
$532$	0	0
$533$	5.17681e19	0.183488
$534$	0	0
$535$	−4.46210e19	−0.153775
$536$	0	0
$537$	8.77717e19	0.294135
$538$	0	0
$539$	3.53801e19	0.115304
$540$	0	0
$541$	1.39993e20	0.443739	0.221870	−	0.975076i	$-0.428784\pi$
$541$	1.39993e20	0.443739	0.221870	+	0.975076i	$0.428784\pi$
$542$	0	0
$543$	1.01728e20	0.313647
$544$	0	0
$545$	−1.42127e20	−0.426288
$546$	0	0
$547$	−2.11459e20	−0.617050	−0.308525	−	0.951216i	$-0.599835\pi$
$547$	−2.11459e20	−0.617050	−0.308525	+	0.951216i	$0.599835\pi$
$548$	0	0
$549$	1.68466e20	0.478320
$550$	0	0
$551$	−2.74252e20	−0.757726
$552$	0	0
$553$	−1.38300e19	−0.0371864
$554$	0	0
$555$	4.85682e19	0.127103
$556$	0	0
$557$	−7.95801e19	−0.202717	−0.101359	−	0.994850i	$-0.532319\pi$
$557$	−7.95801e19	−0.202717	−0.101359	+	0.994850i	$0.532319\pi$
$558$	0	0
$559$	−2.76645e19	−0.0686015
$560$	0	0
$561$	−1.04295e20	−0.251791
$562$	0	0
$563$	−2.32220e20	−0.545867	−0.272934	−	0.962033i	$-0.587994\pi$
$563$	−2.32220e20	−0.545867	−0.272934	+	0.962033i	$0.587994\pi$
$564$	0	0
$565$	5.16770e19	0.118286
$566$	0	0
$567$	−1.88400e19	−0.0419961
$568$	0	0
$569$	−3.60883e20	−0.783474	−0.391737	−	0.920077i	$-0.628126\pi$
$569$	−3.60883e20	−0.783474	−0.391737	+	0.920077i	$0.628126\pi$
$570$	0	0
$571$	9.37473e19	0.198238	0.0991192	−	0.995076i	$-0.468397\pi$
$571$	9.37473e19	0.198238	0.0991192	+	0.995076i	$0.468397\pi$
$572$	0	0
$573$	1.04401e20	0.215052
$574$	0	0
$575$	5.63856e20	1.13151
$576$	0	0
$577$	6.43381e20	1.25791	0.628955	−	0.777442i	$-0.283483\pi$
$577$	6.43381e20	1.25791	0.628955	+	0.777442i	$0.283483\pi$
$578$	0	0
$579$	4.88517e20	0.930658
$580$	0	0
$581$	−5.84685e19	−0.108543
$582$	0	0
$583$	−2.88578e20	−0.522094
$584$	0	0
$585$	−1.23432e19	−0.0217650
$586$	0	0
$587$	−7.78701e20	−1.33840	−0.669199	−	0.743083i	$-0.733363\pi$
$587$	−7.78701e20	−1.33840	−0.669199	+	0.743083i	$0.733363\pi$
$588$	0	0
$589$	9.25272e20	1.55026
$590$	0	0
$591$	−5.83715e20	−0.953443
$592$	0	0
$593$	7.97620e20	1.27024	0.635120	−	0.772413i	$-0.280950\pi$
$593$	7.97620e20	1.27024	0.635120	+	0.772413i	$0.280950\pi$
$594$	0	0
$595$	−3.56092e19	−0.0552949
$596$	0	0
$597$	4.05701e20	0.614325
$598$	0	0
$599$	−4.44026e20	−0.655703	−0.327852	−	0.944729i	$-0.606325\pi$
$599$	−4.44026e20	−0.655703	−0.327852	+	0.944729i	$0.606325\pi$
$600$	0	0
$601$	2.76193e18	0.00397790	0.00198895	−	0.999998i	$-0.499367\pi$
$601$	2.76193e18	0.00397790	0.00198895	+	0.999998i	$0.499367\pi$
$602$	0	0
$603$	−1.07913e20	−0.151598
$604$	0	0
$605$	6.88656e19	0.0943707
$606$	0	0
$607$	5.32399e20	0.711741	0.355871	−	0.934535i	$-0.384184\pi$
$607$	5.32399e20	0.711741	0.355871	+	0.934535i	$0.384184\pi$
$608$	0	0
$609$	7.71575e19	0.100635
$610$	0	0
$611$	3.19348e20	0.406401
$612$	0	0
$613$	4.11517e20	0.511016	0.255508	−	0.966807i	$-0.417757\pi$
$613$	4.11517e20	0.511016	0.255508	+	0.966807i	$0.417757\pi$
$614$	0	0
$615$	−9.81475e19	−0.118937
$616$	0	0
$617$	2.77302e19	0.0327955	0.0163977	−	0.999866i	$-0.494780\pi$
$617$	2.77302e19	0.0327955	0.0163977	+	0.999866i	$0.494780\pi$
$618$	0	0
$619$	4.44060e20	0.512581	0.256290	−	0.966600i	$-0.417500\pi$
$619$	4.44060e20	0.512581	0.256290	+	0.966600i	$0.417500\pi$
$620$	0	0
$621$	−2.08559e20	−0.234986
$622$	0	0
$623$	4.29588e20	0.472489
$624$	0	0
$625$	7.31509e20	0.785451
$626$	0	0
$627$	7.30367e20	0.765658
$628$	0	0
$629$	4.29221e20	0.439340
$630$	0	0
$631$	1.08535e18	0.00108480	0.000542399	−	1.00000i	$-0.499827\pi$
$631$	1.08535e18	0.00108480	0.000542399		1.00000i	$0.499827\pi$
$632$	0	0
$633$	1.03117e21	1.00648
$634$	0	0
$635$	−4.97704e20	−0.474426
$636$	0	0
$637$	3.70044e19	0.0344515
$638$	0	0
$639$	3.95805e20	0.359935
$640$	0	0
$641$	1.73841e21	1.54425	0.772123	−	0.635473i	$-0.219195\pi$
$641$	1.73841e21	1.54425	0.772123	+	0.635473i	$0.219195\pi$
$642$	0	0
$643$	−1.80265e21	−1.56434	−0.782168	−	0.623068i	$-0.785886\pi$
$643$	−1.80265e21	−1.56434	−0.782168	+	0.623068i	$0.785886\pi$
$644$	0	0
$645$	5.24493e19	0.0444674
$646$	0	0
$647$	−9.28108e20	−0.768805	−0.384403	−	0.923166i	$-0.625593\pi$
$647$	−9.28108e20	−0.768805	−0.384403	+	0.923166i	$0.625593\pi$
$648$	0	0
$649$	1.59159e20	0.128823
$650$	0	0
$651$	−2.60314e20	−0.205893
$652$	0	0
$653$	2.13195e21	1.64789	0.823945	−	0.566669i	$-0.191768\pi$
$653$	2.13195e21	1.64789	0.823945	+	0.566669i	$0.191768\pi$
$654$	0	0
$655$	−4.21418e20	−0.318349
$656$	0	0
$657$	6.58616e19	0.0486286
$658$	0	0
$659$	−2.49002e21	−1.79706	−0.898531	−	0.438910i	$-0.855365\pi$
$659$	−2.49002e21	−1.79706	−0.898531	+	0.438910i	$0.855365\pi$
$660$	0	0
$661$	−1.45542e21	−1.02678	−0.513389	−	0.858156i	$-0.671610\pi$
$661$	−1.45542e21	−1.02678	−0.513389	+	0.858156i	$0.671610\pi$
$662$	0	0
$663$	−1.09083e20	−0.0752324
$664$	0	0
$665$	2.49369e20	0.168143
$666$	0	0
$667$	8.54133e20	0.563095
$668$	0	0
$669$	−1.18255e21	−0.762293
$670$	0	0
$671$	1.83738e21	1.15819
$672$	0	0
$673$	2.12283e21	1.30859	0.654294	−	0.756241i	$-0.272966\pi$
$673$	2.12283e21	1.30859	0.654294	+	0.756241i	$0.272966\pi$
$674$	0	0
$675$	−2.95823e20	−0.178342
$676$	0	0
$677$	3.30291e21	1.94752	0.973759	−	0.227582i	$-0.0730819\pi$
$677$	3.30291e21	1.94752	0.973759	+	0.227582i	$0.0730819\pi$
$678$	0	0
$679$	−7.53907e18	−0.00434805
$680$	0	0
$681$	−5.03533e20	−0.284070
$682$	0	0
$683$	−1.53723e21	−0.848369	−0.424184	−	0.905576i	$-0.639439\pi$
$683$	−1.53723e21	−0.848369	−0.424184	+	0.905576i	$0.639439\pi$
$684$	0	0
$685$	3.33291e20	0.179947
$686$	0	0
$687$	3.63521e17	0.000192023 0
$688$	0	0
$689$	−3.01826e20	−0.155996
$690$	0	0
$691$	−5.83157e20	−0.294917	−0.147459	−	0.989068i	$-0.547109\pi$
$691$	−5.83157e20	−0.294917	−0.147459	+	0.989068i	$0.547109\pi$
$692$	0	0
$693$	−2.05480e20	−0.101688
$694$	0	0
$695$	−8.18649e20	−0.396471
$696$	0	0
$697$	−8.67377e20	−0.411114
$698$	0	0
$699$	−1.41339e21	−0.655668
$700$	0	0
$701$	−1.20373e21	−0.546569	−0.273285	−	0.961933i	$-0.588110\pi$
$701$	−1.20373e21	−0.546569	−0.273285	+	0.961933i	$0.588110\pi$
$702$	0	0
$703$	−3.00580e21	−1.33597
$704$	0	0
$705$	−6.05454e20	−0.263429
$706$	0	0
$707$	−1.19017e21	−0.506947
$708$	0	0
$709$	−7.91394e20	−0.330024	−0.165012	−	0.986292i	$-0.552766\pi$
$709$	−7.91394e20	−0.330024	−0.165012	+	0.986292i	$0.552766\pi$
$710$	0	0
$711$	8.03219e19	0.0327953
$712$	0	0
$713$	−2.88168e21	−1.15206
$714$	0	0
$715$	−1.34622e20	−0.0527013
$716$	0	0
$717$	4.01125e20	0.153775
$718$	0	0
$719$	1.29628e19	0.00486669	0.00243334	−	0.999997i	$-0.499225\pi$
$719$	1.29628e19	0.00486669	0.00243334	+	0.999997i	$0.499225\pi$
$720$	0	0
$721$	4.28475e20	0.157547
$722$	0	0
$723$	−2.01555e21	−0.725865
$724$	0	0
$725$	1.21152e21	0.427360
$726$	0	0
$727$	−3.09572e21	−1.06968	−0.534840	−	0.844954i	$-0.679628\pi$
$727$	−3.09572e21	−1.06968	−0.534840	+	0.844954i	$0.679628\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	4.63520e20	0.153705
$732$	0	0
$733$	3.91138e19	0.0127072	0.00635360	−	0.999980i	$-0.497978\pi$
$733$	3.91138e19	0.0127072	0.00635360	+	0.999980i	$0.497978\pi$
$734$	0	0
$735$	−7.01569e19	−0.0223314
$736$	0	0
$737$	−1.17696e21	−0.367076
$738$	0	0
$739$	3.67717e21	1.12378	0.561888	−	0.827213i	$-0.310075\pi$
$739$	3.67717e21	1.12378	0.561888	+	0.827213i	$0.310075\pi$
$740$	0	0
$741$	7.63899e20	0.228770
$742$	0	0
$743$	2.72503e21	0.799753	0.399877	−	0.916569i	$-0.369053\pi$
$743$	2.72503e21	0.799753	0.399877	+	0.916569i	$0.369053\pi$
$744$	0	0
$745$	−1.77625e21	−0.510894
$746$	0	0
$747$	3.39573e20	0.0957257
$748$	0	0
$749$	−7.76919e20	−0.214666
$750$	0	0
$751$	−6.29535e21	−1.70498	−0.852492	−	0.522740i	$-0.824910\pi$
$751$	−6.29535e21	−1.70498	−0.852492	+	0.522740i	$0.824910\pi$
$752$	0	0
$753$	−1.81382e21	−0.481541
$754$	0	0
$755$	−8.42071e20	−0.219153
$756$	0	0
$757$	−1.30427e21	−0.332773	−0.166387	−	0.986061i	$-0.553210\pi$
$757$	−1.30427e21	−0.332773	−0.166387	+	0.986061i	$0.553210\pi$
$758$	0	0
$759$	−2.27467e21	−0.568989
$760$	0	0
$761$	1.12946e19	0.00277005	0.00138502	−	0.999999i	$-0.499559\pi$
$761$	1.12946e19	0.00277005	0.00138502	+	0.999999i	$0.499559\pi$
$762$	0	0
$763$	−2.47465e21	−0.595086
$764$	0	0
$765$	2.06811e20	0.0487655
$766$	0	0
$767$	1.66466e20	0.0384910
$768$	0	0
$769$	−1.32243e21	−0.299864	−0.149932	−	0.988696i	$-0.547905\pi$
$769$	−1.32243e21	−0.299864	−0.149932	+	0.988696i	$0.547905\pi$
$770$	0	0
$771$	−4.54718e21	−1.01119
$772$	0	0
$773$	3.16287e21	0.689818	0.344909	−	0.938636i	$-0.387910\pi$
$773$	3.16287e21	0.689818	0.344909	+	0.938636i	$0.387910\pi$
$774$	0	0
$775$	−4.08742e21	−0.874352
$776$	0	0
$777$	8.45647e20	0.177432
$778$	0	0
$779$	6.07418e21	1.25013
$780$	0	0
$781$	4.31688e21	0.871539
$782$	0	0
$783$	−4.48115e20	−0.0887516
$784$	0	0
$785$	1.52878e20	0.0297046
$786$	0	0
$787$	−6.78002e20	−0.129247	−0.0646235	−	0.997910i	$-0.520585\pi$
$787$	−6.78002e20	−0.129247	−0.0646235	+	0.997910i	$0.520585\pi$
$788$	0	0
$789$	−2.99249e20	−0.0559701
$790$	0	0
$791$	8.99776e20	0.165124
$792$	0	0
$793$	1.92174e21	0.346055
$794$	0	0
$795$	5.72235e20	0.101116
$796$	0	0
$797$	1.32682e20	0.0230077	0.0115039	−	0.999934i	$-0.496338\pi$
$797$	1.32682e20	0.0230077	0.0115039	+	0.999934i	$0.496338\pi$
$798$	0	0
$799$	−5.35069e21	−0.910560
$800$	0	0
$801$	−2.49496e21	−0.416696
$802$	0	0
$803$	7.18325e20	0.117748
$804$	0	0
$805$	−7.76637e20	−0.124954
$806$	0	0
$807$	4.39563e21	0.694175
$808$	0	0
$809$	8.60534e21	1.33400	0.666998	−	0.745060i	$-0.267579\pi$
$809$	8.60534e21	1.33400	0.666998	+	0.745060i	$0.267579\pi$
$810$	0	0
$811$	−2.80971e21	−0.427568	−0.213784	−	0.976881i	$-0.568579\pi$
$811$	−2.80971e21	−0.427568	−0.213784	+	0.976881i	$0.568579\pi$
$812$	0	0
$813$	−2.70412e21	−0.403969
$814$	0	0
$815$	−6.34195e20	−0.0930124
$816$	0	0
$817$	−3.24599e21	−0.467393
$818$	0	0
$819$	−2.14914e20	−0.0303833
$820$	0	0
$821$	6.29128e21	0.873304	0.436652	−	0.899631i	$-0.356164\pi$
$821$	6.29128e21	0.873304	0.436652	+	0.899631i	$0.356164\pi$
$822$	0	0
$823$	7.58806e21	1.03427	0.517133	−	0.855905i	$-0.326999\pi$
$823$	7.58806e21	1.03427	0.517133	+	0.855905i	$0.326999\pi$
$824$	0	0
$825$	−3.22642e21	−0.431833
$826$	0	0
$827$	7.06920e21	0.929136	0.464568	−	0.885538i	$-0.346210\pi$
$827$	7.06920e21	0.929136	0.464568	+	0.885538i	$0.346210\pi$
$828$	0	0
$829$	−8.55256e20	−0.110392	−0.0551959	−	0.998476i	$-0.517578\pi$
$829$	−8.55256e20	−0.110392	−0.0551959	+	0.998476i	$0.517578\pi$
$830$	0	0
$831$	2.47290e21	0.313472
$832$	0	0
$833$	−6.20011e20	−0.0771901
$834$	0	0
$835$	−5.72914e20	−0.0700553
$836$	0	0
$837$	1.51185e21	0.181580
$838$	0	0
$839$	2.80175e21	0.330533	0.165267	−	0.986249i	$-0.447152\pi$
$839$	2.80175e21	0.330533	0.165267	+	0.986249i	$0.447152\pi$
$840$	0	0
$841$	−6.79398e21	−0.787325
$842$	0	0
$843$	−1.29488e21	−0.147409
$844$	0	0
$845$	2.28022e21	0.255007
$846$	0	0
$847$	1.19906e21	0.131739
$848$	0	0
$849$	−5.85717e21	−0.632238
$850$	0	0
$851$	9.36131e21	0.992807
$852$	0	0
$853$	7.68392e21	0.800691	0.400345	−	0.916364i	$-0.368890\pi$
$853$	7.68392e21	0.800691	0.400345	+	0.916364i	$0.368890\pi$
$854$	0	0
$855$	−1.44828e21	−0.148288
$856$	0	0
$857$	5.67734e21	0.571201	0.285601	−	0.958349i	$-0.407807\pi$
$857$	5.67734e21	0.571201	0.285601	+	0.958349i	$0.407807\pi$
$858$	0	0
$859$	1.86296e22	1.84186	0.920928	−	0.389734i	$-0.127433\pi$
$859$	1.86296e22	1.84186	0.920928	+	0.389734i	$0.127433\pi$
$860$	0	0
$861$	−1.70890e21	−0.166032
$862$	0	0
$863$	−7.71816e21	−0.736941	−0.368471	−	0.929639i	$-0.620118\pi$
$863$	−7.71816e21	−0.736941	−0.368471	+	0.929639i	$0.620118\pi$
$864$	0	0
$865$	−3.89925e21	−0.365898
$866$	0	0
$867$	−4.43243e21	−0.408789
$868$	0	0
$869$	8.76037e20	0.0794098
$870$	0	0
$871$	−1.23100e21	−0.109678
$872$	0	0
$873$	4.37854e19	0.00383462
$874$	0	0
$875$	−2.29033e21	−0.197169
$876$	0	0
$877$	−2.27791e21	−0.192770	−0.0963848	−	0.995344i	$-0.530728\pi$
$877$	−2.27791e21	−0.192770	−0.0963848	+	0.995344i	$0.530728\pi$
$878$	0	0
$879$	−4.22793e21	−0.351731
$880$	0	0
$881$	8.05842e21	0.659068	0.329534	−	0.944144i	$-0.393108\pi$
$881$	8.05842e21	0.659068	0.329534	+	0.944144i	$0.393108\pi$
$882$	0	0
$883$	−1.63863e22	−1.31757	−0.658787	−	0.752330i	$-0.728930\pi$
$883$	−1.63863e22	−1.31757	−0.658787	+	0.752330i	$0.728930\pi$
$884$	0	0
$885$	−3.15604e20	−0.0249498
$886$	0	0
$887$	1.30780e22	1.01652	0.508258	−	0.861205i	$-0.330290\pi$
$887$	1.30780e22	1.01652	0.508258	+	0.861205i	$0.330290\pi$
$888$	0	0
$889$	−8.66579e21	−0.662285
$890$	0	0
$891$	1.19339e21	0.0896806
$892$	0	0
$893$	3.74705e22	2.76887
$894$	0	0
$895$	−1.89826e21	−0.137938
$896$	0	0
$897$	−2.37910e21	−0.170008
$898$	0	0
$899$	−6.19165e21	−0.435120
$900$	0	0
$901$	5.05712e21	0.349516
$902$	0	0
$903$	9.13223e20	0.0620753
$904$	0	0
$905$	−2.20009e21	−0.147088
$906$	0	0
$907$	9.17057e21	0.603034	0.301517	−	0.953461i	$-0.402507\pi$
$907$	9.17057e21	0.603034	0.301517	+	0.953461i	$0.402507\pi$
$908$	0	0
$909$	6.91224e21	0.447085
$910$	0	0
$911$	−2.86426e21	−0.182232	−0.0911159	−	0.995840i	$-0.529043\pi$
$911$	−2.86426e21	−0.182232	−0.0911159	+	0.995840i	$0.529043\pi$
$912$	0	0
$913$	3.70358e21	0.231788
$914$	0	0
$915$	−3.64344e21	−0.224313
$916$	0	0
$917$	−7.33753e21	−0.444406
$918$	0	0
$919$	2.69345e22	1.60488	0.802439	−	0.596734i	$-0.203535\pi$
$919$	2.69345e22	1.60488	0.802439	+	0.596734i	$0.203535\pi$
$920$	0	0
$921$	1.81502e22	1.06398
$922$	0	0
$923$	4.51507e21	0.260406
$924$	0	0
$925$	1.32782e22	0.753489
$926$	0	0
$927$	−2.48849e21	−0.138944
$928$	0	0
$929$	−4.15281e21	−0.228152	−0.114076	−	0.993472i	$-0.536391\pi$
$929$	−4.15281e21	−0.228152	−0.114076	+	0.993472i	$0.536391\pi$
$930$	0	0
$931$	4.34189e21	0.234723
$932$	0	0
$933$	−2.68821e21	−0.143005
$934$	0	0
$935$	2.25560e21	0.118080
$936$	0	0
$937$	−1.14193e22	−0.588292	−0.294146	−	0.955761i	$-0.595035\pi$
$937$	−1.14193e22	−0.588292	−0.294146	+	0.955761i	$0.595035\pi$
$938$	0	0
$939$	1.17528e21	0.0595866
$940$	0	0
$941$	−2.72313e22	−1.35877	−0.679386	−	0.733781i	$-0.737754\pi$
$941$	−2.72313e22	−1.35877	−0.679386	+	0.733781i	$0.737754\pi$
$942$	0	0
$943$	−1.89175e22	−0.929022
$944$	0	0
$945$	4.07457e20	0.0196944
$946$	0	0
$947$	2.51727e22	1.19758	0.598790	−	0.800906i	$-0.295648\pi$
$947$	2.51727e22	1.19758	0.598790	+	0.800906i	$0.295648\pi$
$948$	0	0
$949$	7.51304e20	0.0351819
$950$	0	0
$951$	5.11214e21	0.235640
$952$	0	0
$953$	−3.72120e22	−1.68844	−0.844222	−	0.535994i	$-0.819937\pi$
$953$	−3.72120e22	−1.68844	−0.844222	+	0.535994i	$0.819937\pi$
$954$	0	0
$955$	−2.25789e21	−0.100851
$956$	0	0
$957$	−4.88740e21	−0.214901
$958$	0	0
$959$	5.80310e21	0.251200
$960$	0	0
$961$	−2.57584e21	−0.109772
$962$	0	0
$963$	4.51219e21	0.189317
$964$	0	0
$965$	−1.05653e22	−0.436440
$966$	0	0
$967$	2.32377e22	0.945138	0.472569	−	0.881294i	$-0.343327\pi$
$967$	2.32377e22	0.945138	0.472569	+	0.881294i	$0.343327\pi$
$968$	0	0
$969$	−1.27992e22	−0.512570
$970$	0	0
$971$	2.52851e22	0.997059	0.498530	−	0.866873i	$-0.333874\pi$
$971$	2.52851e22	0.997059	0.498530	+	0.866873i	$0.333874\pi$
$972$	0	0
$973$	−1.42539e22	−0.553463
$974$	0	0
$975$	−3.37455e21	−0.129027
$976$	0	0
$977$	2.06700e22	0.778272	0.389136	−	0.921180i	$-0.372774\pi$
$977$	2.06700e22	0.778272	0.389136	+	0.921180i	$0.372774\pi$
$978$	0	0
$979$	−2.72115e22	−1.00898
$980$	0	0
$981$	1.43723e22	0.524816
$982$	0	0
$983$	−1.25066e22	−0.449767	−0.224883	−	0.974386i	$-0.572200\pi$
$983$	−1.25066e22	−0.449767	−0.224883	+	0.974386i	$0.572200\pi$
$984$	0	0
$985$	1.26241e22	0.447126
$986$	0	0
$987$	−1.05419e22	−0.367739
$988$	0	0
$989$	1.01094e22	0.347338
$990$	0	0
$991$	4.41671e22	1.49467	0.747337	−	0.664445i	$-0.231332\pi$
$991$	4.41671e22	1.49467	0.747337	+	0.664445i	$0.231332\pi$
$992$	0	0
$993$	−1.34756e22	−0.449187
$994$	0	0
$995$	−8.77418e21	−0.288093
$996$	0	0
$997$	1.26229e22	0.408268	0.204134	−	0.978943i	$-0.434562\pi$
$997$	1.26229e22	0.408268	0.204134	+	0.978943i	$0.434562\pi$
$998$	0	0
$999$	−4.91135e21	−0.156480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.16.a.d.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.16.a.d.1.2	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q+2187.00 q^{3} -47298.7 q^{5} -823543. q^{7} +4.78297e6 q^{9} +O(q^{10})\)	\(q+2187.00 q^{3} -47298.7 q^{5} -823543. q^{7} +4.78297e6 q^{9} +5.21658e7 q^{11} +5.45608e7 q^{13} -1.03442e8 q^{15} -9.14169e8 q^{17} +6.40186e9 q^{19} -1.80109e9 q^{21} -1.99380e10 q^{23} -2.82804e10 q^{25} +1.04604e10 q^{27} -4.28394e10 q^{29} +1.44532e11 q^{31} +1.14087e11 q^{33} +3.89525e10 q^{35} -4.69520e11 q^{37} +1.19324e11 q^{39} +9.48815e11 q^{41} -5.07039e11 q^{43} -2.26228e11 q^{45} +5.85306e12 q^{47} +6.78223e11 q^{49} -1.99929e12 q^{51} -5.53193e12 q^{53} -2.46738e12 q^{55} +1.40009e13 q^{57} +3.05101e12 q^{59} +3.52220e13 q^{61} -3.93898e12 q^{63} -2.58066e12 q^{65} -2.25619e13 q^{67} -4.36045e13 q^{69} +8.27529e13 q^{71} +1.37700e13 q^{73} -6.18493e13 q^{75} -4.29608e13 q^{77} +1.67933e13 q^{79} +2.28768e13 q^{81} +7.09963e13 q^{83} +4.32390e13 q^{85} -9.36897e13 q^{87} -5.21634e14 q^{89} -4.49332e13 q^{91} +3.16091e14 q^{93} -3.02800e14 q^{95} +9.15443e12 q^{97} +2.49508e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9}+O(q^{10}) \) 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9	\( 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9} - 61540668 q^{11} + 369006456 q^{13} + 602378532 q^{15} - 1458569484 q^{17} - 2691246184 q^{19} - 7204354164 q^{21} + 18028282212 q^{23} + 8448695516 q^{25} + 41841412812 q^{27} - 14387019264 q^{29} + 271402618248 q^{31} - 134589440916 q^{33} - 226833389748 q^{35} + 1719450823976 q^{37} + 807017119272 q^{39} + 2784989291868 q^{41} + 1897932676640 q^{43} + 1317401849484 q^{45} + 1872077464008 q^{47} + 2712892291396 q^{49} - 3189891461508 q^{51} + 4740642191736 q^{53} - 21671359654504 q^{55} - 5885755404408 q^{57} + 13984487218296 q^{59} + 32810552663568 q^{61} - 15755922556668 q^{63} + 183566971857672 q^{65} + 4158516498856 q^{67} + 39427853197644 q^{69} + 70250046828444 q^{71} + 27064251481456 q^{73} + 18477297093492 q^{75} + 50681386346724 q^{77} - 238381767283992 q^{79} + 91507169819844 q^{81} - 233193205883808 q^{83} - 893307076687696 q^{85} - 31464411130368 q^{87} + 316154161239228 q^{89} - 303892683793608 q^{91} + 593557526108376 q^{93} - 11\!\cdots\!76 q^{95}+ \cdots - 294347107283292 q^{99}+O(q^{100}) \) 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9 - 61540668 * q^11 + 369006456 * q^13 + 602378532 * q^15 - 1458569484 * q^17 - 2691246184 * q^19 - 7204354164 * q^21 + 18028282212 * q^23 + 8448695516 * q^25 + 41841412812 * q^27 - 14387019264 * q^29 + 271402618248 * q^31 - 134589440916 * q^33 - 226833389748 * q^35 + 1719450823976 * q^37 + 807017119272 * q^39 + 2784989291868 * q^41 + 1897932676640 * q^43 + 1317401849484 * q^45 + 1872077464008 * q^47 + 2712892291396 * q^49 - 3189891461508 * q^51 + 4740642191736 * q^53 - 21671359654504 * q^55 - 5885755404408 * q^57 + 13984487218296 * q^59 + 32810552663568 * q^61 - 15755922556668 * q^63 + 183566971857672 * q^65 + 4158516498856 * q^67 + 39427853197644 * q^69 + 70250046828444 * q^71 + 27064251481456 * q^73 + 18477297093492 * q^75 + 50681386346724 * q^77 - 238381767283992 * q^79 + 91507169819844 * q^81 - 233193205883808 * q^83 - 893307076687696 * q^85 - 31464411130368 * q^87 + 316154161239228 * q^89 - 303892683793608 * q^91 + 593557526108376 * q^93 - 1197067113655776 * q^95 + 529302182472448 * q^97 - 294347107283292 * q^99

Introduction

L-functions

Modular forms

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Groups

Database

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